I believe that this exercise can be resolve using Morera's Thm. I multiplied the expression by the denominator function of $f$ but it did not help.
Could anybody help me on this one?
Let $D$ a connected open set. Assume that $f$ is the quotient of two analytic functions on $D$. Show that if $a_{0}(z) + a_{1}(z)f(z) + a_{2}(z)(f(z))^{2}+ ... + a_{n-1}(z)(f(z))^{n-1}+(f(z))^{n}=0$ for some analytic functions $a_{0}(z),a_{1}(z),a_{2}(z),... , a_{n-1}(z)$ then f is analytic.
Let $f=\frac g h$ with g and h analytic. We may suppose that g is not a constant so its zeros are isolated. We may also assume that p and q have no common zeros. Let $h(c)=0$ and let $N$ be the order of the zero. Multiply the given equation by $(z-c)^{Nn}$. Letting $z \to c$ we find that $g(c)=0$. This contradicts our assumption that g and h have no common zeros. We have proved that h cannot have any zeros after all so f is analytic.